Introduction to Reconfiguration

Nishimura, Nobuya

doi:10.3390/a11040052

Cited by 189 publications

(123 citation statements)

References 129 publications

(171 reference statements)

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“…The question is whether S can be transformed to T by repeated applications of the modification rule in a way that maintains the solution feasible at all times. Due to their numerous applications, reconfiguration problems have attracted much interest in the literature, and reconfiguration versions of standard problems (such as Satisfiability, Dominating Set, and Independent Set) have been widely studied (see the surveys [10,19] and the references therein). Among reconfiguration problems on graphs, Independent Set Reconfiguration is certainly the most well-studied.…”

Section: Introductionmentioning

confidence: 99%

Token Sliding on Split Graphs

et al. 2020

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We consider the complexity of the Independent Set Reconfiguration problem under the Token Sliding rule. In this problem we are given two independent sets of a graph and are asked if we can transform one to the other by repeatedly exchanging a vertex that is currently in the set with one of its neighbors, while maintaining the set independent. Our main result is to show that this problem is PSPACE-complete on split graphs (and hence also on chordal graphs), thus resolving an open problem in this area.We then go on to consider the c-Colorable Reconfiguration problem under the same rule, where the constraint is now to maintain the set c-colorable at all times. As one may expect, a simple modification of our reduction shows that this more general problem is PSPACE-complete for all fixed c ≥ 1 on chordal graphs. Somewhat surprisingly, we show that the same cannot be said for split graphs: we give a polynomial time (n O(c) ) algorithm for all fixed values of c, except c = 1, for which the problem is PSPACE-complete. We complement our algorithm with a lower bound showing that c-Colorable Reconfiguration is W[2]-hard on split graphs parameterized by c and the length of the solution, as well as a tight ETH-based lower bound for both parameters.

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Section: Introductionmentioning

confidence: 99%

Token Sliding on Split Graphs

et al. 2020

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“…Another focus is to determine the diameter of the reconfiguration graph in case it is connected or the diameter of its components if it is disconnected [3,7,2,5,11]. We refer the reader to [15,13] for excellent surveys on reconfiguration problems.…”

Section: Introductionmentioning

confidence: 99%

Reconfiguration graph for vertex colourings of weakly chordal graphs

Feghali

Fiala

2020

Discrete Mathematics

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The reconfiguration graph R k (G) of the k-colourings of a graph G contains as its vertex set the k-colourings of G and two colourings are joined by an edge if they differ in colour on just one vertex of G.We show that for each k ≥ 3 there is a k-colourable weakly chordal graph G such that R k+1 (G) is disconnected. We also introduce a subclass of k-colourable weakly chordal graphs which we call k-colourable compact graphs and show that for each k-colourable compact graph G on n vertices, R k+1 (G) has diameter O(n 2 ). We show that this class contains all k-colourable co-chordal graphs and when k = 3 all 3-colourable (P 5 , P 5 , C 5 )free graphs. We also mention some open problems. *

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“…The question is whether one can transform an arbitrary configuration to the one where each face of the cube has only one color. For an overview of this research area, readers are referred to the recent surveys by van den Heuvel [16] and Nishimura [22].…”

Section: Introductionmentioning

confidence: 99%

“…The Vertex Cover Reconfiguration (VCR) problem is one of the most well-studied reconfiguration problems, from both classical and parameterized complexity viewpoints (e.g., see [22] for a quick summary of known results). It is well-known that if I is a vertex cover of a graph G = (V, E) then V \I is an independent set of G, i.e., a vertex-subset whose members are pairwise non-adjacent.…”

Section: Introductionmentioning

confidence: 99%

Reconfiguring k-path Vertex Covers

Hoang

Suzuki

Yagita

2020

WALCOM: Algorithms and Computation

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A vertex subset I of a graph G is called a k-path vertex cover if every path on k vertices in G contains at least one vertex from I. The k-Path Vertex Cover Reconfiguration (k-PVCR) problem asks if one can transform one k-path vertex cover into another via a sequence of k-path vertex covers where each intermediate member is obtained from its predecessor by applying a given reconfiguration rule exactly once. We investigate the computational complexity of k-PVCR from the viewpoint of graph classes under the well-known reconfiguration rules: TS, TJ, and TAR. The problem for k = 2, known as the Vertex Cover Reconfiguration (VCR) problem, has been well-studied in the literature. We show that certain known hardness results for VCR on different graph classes including planar graphs, bounded bandwidth graphs, chordal graphs, and bipartite graphs, can be extended for k-PVCR. In particular, we prove a complexity dichotomy for k-PVCR on general graphs: on those whose maximum degree is 3 (and even planar), the problem is PSPACE-complete, while on those whose maximum degree is 2 (i.e., paths and cycles), the problem can be solved in polynomial time. Additionally, we also design polynomial-time algorithms for k-PVCR on trees under each of TJ and TAR. Moreover, on paths, cycles, and trees, we describe how one can construct a reconfiguration sequence between two given k-path vertex covers in a yes-instance. In particular, on paths, our constructed reconfiguration sequence is shortest.

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Introduction to Reconfiguration

Cited by 189 publications

References 129 publications

Token Sliding on Split Graphs

Token Sliding on Split Graphs

Reconfiguration graph for vertex colourings of weakly chordal graphs

Reconfiguring k-path Vertex Covers

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